Asymptotes of (x+y)²(x+2y+2)=x+9y−2

Curve: (x + y)²(x + 2y + 2) = x + 9y − 2, i.e. (x+y)²(x+2y+2) − x − 9y + 2 = 0. Degree 3. Recall: slopes from φ₃(m) = 0; simple root: c = −φ₂(m)/φ₃′(m); double root: c² φ₃″/2 + c φ₂′ + φ₁ = 0.

φ₃(m) = (1+m)²(1+2m)
φ₂(m) = 2(1+m)²
φ₁(m) = −(1 + 9m)
φ₀    = 2

φ₃(m) = 0 gives m = −1 (twice) and m = −1/2 (once).

m = −1/2 (simple): φ₃′(m) = 2(1+m)(1+2m) + 2(1+m)² = 0 + 2(1/4) = 1/2 φ₂(−1/2) = 2(1/4) = 1/2 c = −(1/2)/(1/2) = −1

Asymptote: y = −x/2 − 1, i.e. x + 2y + 2 = 0.

m = −1 (double): φ₃″(m)/2 at m = −1 equals (1 + 2m) = −1; φ₂′(−1) = 4(1+m) = 0; φ₁(−1) = −(1 − 9) = 8.

−c² + 0 + 8 = 0, so c = ±2√2.

Asymptotes: y = −x + 2√2 and y = −x − 2√2, i.e. x + y = ±2√2.

Answer: x + 2y + 2 = 0, x + y − 2√2 = 0, x + y + 2√2 = 0.